June  2020, 13(3): 623-651. doi: 10.3934/krm.2020021

Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes equations in the whole domain

1. 

Institute of New Media and Communications, Seoul National University, Seoul 08826, Republic of Korea

2. 

College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

* Corresponding author: Weiyuan Zou

Received  December 2019 Revised  February 2020 Published  March 2020

Fund Project: The second author Weiyuan Zou is supported by the Fundamental Research Funds for the Central Universities ZY1937

We study local existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker–Smale (in short TCS) model coupled with incompressible Navier–Stokes (NS) equations in the whole space. The coupled system consists of the kinetic TCS equation for particle ensemble and the incompressible NS equations for a fluid via a drag force. For the strong solution, we investigate the blow-up mechanism for the coupled system, and we also study the global existence of a weak solution in the whole space.

Citation: Jeongho Kim, Weiyuan Zou. Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes equations in the whole domain. Kinetic & Related Models, 2020, 13 (3) : 623-651. doi: 10.3934/krm.2020021
References:
[1]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Time-asymptotic interaction of flocking particles and incompressible viscous fluid,, Nonlinearity, 25 (2012), 1155-1177.  doi: 10.1088/0951-7715/25/4/1155.  Google Scholar

[2]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids, Discrete and Continuous Dynamical System, 34 (2014), 4419-4458.  doi: 10.3934/dcds.2014.34.4419.  Google Scholar

[3]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Global existence of strong solution for the Cucker-Smale-Navier-Stokes system, J. Differ. Equat., 257 (2014), 2225-2255.  doi: 10.1016/j.jde.2014.05.035.  Google Scholar

[4]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Global existence of strong solutions to the Cucker-Smale-Stokes system, J. Math. Fluid Mech., 18 (2016), 381-396.  doi: 10.1007/s00021-015-0237-2.  Google Scholar

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C. BarangerL. BoudinP.-E. Jabin and S. Mancini, A modeling of biospray for the upper airways, ESAIM Proc., 14 (2005), 41-47.   Google Scholar

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C. Baranger and L. Desvillettes, Coupling Euler and Vlasov equation in the context of sprays: the local-in-time, classical solutions, J. Hyperbolic Differ. Equ., 3 (2006), 1-26.  doi: 10.1142/S0219891606000707.  Google Scholar

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L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solution for the coupled Vlasov and Navier-Stokes equations, Differential Integral Equations, 22 (2009), 1247-1271.   Google Scholar

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J. A. CarrilloY.-P. Choi and T. K. Karper, On the analysis of a coupled kinetic-fluid model with local alignment forces,, Ann. I. H. Poincare-AN., 33 (2016), 273-307.  doi: 10.1016/j.anihpc.2014.10.002.  Google Scholar

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Y.-P. Choi, Large-time behavior for the Vlasov/compressible Navier-Stokes equations,, J. Math. Phys., 57 (2016), 071501, 13pp. doi: 10.1063/1.4955026.  Google Scholar

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Y.-P. Choi and J. Lee, Global existence of weak and strong solutions to Cucker-Smale-Navier-Stokes equations in $ \mathbb R^2$, Nonlinear Analysis: Real World Applications, 27 (2016), 158-182.  doi: 10.1016/j.nonrwa.2015.07.013.  Google Scholar

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Y.-P. Choi and B. Kwon, Global well-posedness and large-time behavior for the inhomogeneous Vlasov-Navier-Stokes equations, Nonlinearity, 28 (2015), 3309-3336.  doi: 10.1088/0951-7715/28/9/3309.  Google Scholar

[12]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[13]

Y.-P. ChoiS.-Y. HaJ. Jung and J. Kim, Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids, Nonlinearity, 32 (2019), 1597-1640.  doi: 10.1088/1361-6544/aafaae.  Google Scholar

[14]

Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim, On the coupling of kinetic thermomechanical Cucker-Smale equation and compressible viscous fluid system, J. Math. Fluid Mech., 22 (2020), Art. 4, 34 pp. doi: 10.1007/s00021-019-0466-x.  Google Scholar

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J.-G. DongS.-Y. HaD. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.  Google Scholar

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K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations, Japan J. Indust. Appl. Math., 15 (1998), 51-74.  doi: 10.1007/BF03167396.  Google Scholar

[17]

S.-Y. HaJ. KimC. H. MinT. Ruggeri and X. Zhang, A global existence of classical solutions to the hydrodynamic Cucker-Smale model in presence of a temperature field, Anal. Appl., 16 (2018), 757-805.  doi: 10.1142/S0219530518500033.  Google Scholar

[18]

S.-Y. HaJ. KimC. H. MinT. Ruggeri and X. Zhang, Uniform stability and mean-field limit of a thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.  Google Scholar

[19]

S.-Y. HaJ. Kim and T. Ruggeri, Emergent behaviors of thermodynamic Cucker–Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.  doi: 10.1137/17M111064X.  Google Scholar

[20]

S.-Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal., 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.  Google Scholar

[21]

F. LiY. Mu and D. Wang, Strong solutions to the compressible Navier-stokes-Vlasov-Fokker-Plank equations: global existence near the equilibrium and large time behavior, SIAM J. Math. Anal., 49 (2017), 984-1026.  doi: 10.1137/15M1053049.  Google Scholar

[22]

J. Mathiaud, Local smooth solutions of a thin spray model with collisions, Math. Models Methods Appl. Sci., 20 (2010), 191-221.  doi: 10.1142/S0218202510004192.  Google Scholar

[23]

A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Plank/Navier-Stokes system of equations,, Math. Models Methods Appl. Sci., 17 (2007), 1039-1063.  doi: 10.1142/S0218202507002194.  Google Scholar

[24]

P. O'Rourke, Collective Drop Effects on Vaporizing Liquid Sprays, Princeton University, Princeton, NJ, 1981. Google Scholar

[25]

W. E. Ranz and W. R. Marshall, Evaporization from drops, part Ⅰ-Ⅱ, Chem. Eng. Prog., 48 (1952), 141-180.   Google Scholar

[26]

I. VinkovicC. AguirreS. Simoëns and M. Gorokhovski, Large eddy simulation of droplet dispersion for inhomogeneous turbulent wall flow, Int. J. Multiph. Flow, 32 (2006), 344-364.   Google Scholar

[27]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[28]

D. Wang and C. Yu, Global weak solutions to the inhomogeneous Navier-Stokes-Vlasov equations, J. Differential Equations, 259 (2014), 3976-4008.  doi: 10.1016/j.jde.2015.05.016.  Google Scholar

[29]

C. Yu, Global weak solutions to the incompressible Navier-Stokes-Vlasov equations, J. Math. Pures Appl., 100 (2013), 275-293.  doi: 10.1016/j.matpur.2013.01.001.  Google Scholar

show all references

References:
[1]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Time-asymptotic interaction of flocking particles and incompressible viscous fluid,, Nonlinearity, 25 (2012), 1155-1177.  doi: 10.1088/0951-7715/25/4/1155.  Google Scholar

[2]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids, Discrete and Continuous Dynamical System, 34 (2014), 4419-4458.  doi: 10.3934/dcds.2014.34.4419.  Google Scholar

[3]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Global existence of strong solution for the Cucker-Smale-Navier-Stokes system, J. Differ. Equat., 257 (2014), 2225-2255.  doi: 10.1016/j.jde.2014.05.035.  Google Scholar

[4]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Global existence of strong solutions to the Cucker-Smale-Stokes system, J. Math. Fluid Mech., 18 (2016), 381-396.  doi: 10.1007/s00021-015-0237-2.  Google Scholar

[5]

C. BarangerL. BoudinP.-E. Jabin and S. Mancini, A modeling of biospray for the upper airways, ESAIM Proc., 14 (2005), 41-47.   Google Scholar

[6]

C. Baranger and L. Desvillettes, Coupling Euler and Vlasov equation in the context of sprays: the local-in-time, classical solutions, J. Hyperbolic Differ. Equ., 3 (2006), 1-26.  doi: 10.1142/S0219891606000707.  Google Scholar

[7]

L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solution for the coupled Vlasov and Navier-Stokes equations, Differential Integral Equations, 22 (2009), 1247-1271.   Google Scholar

[8]

J. A. CarrilloY.-P. Choi and T. K. Karper, On the analysis of a coupled kinetic-fluid model with local alignment forces,, Ann. I. H. Poincare-AN., 33 (2016), 273-307.  doi: 10.1016/j.anihpc.2014.10.002.  Google Scholar

[9]

Y.-P. Choi, Large-time behavior for the Vlasov/compressible Navier-Stokes equations,, J. Math. Phys., 57 (2016), 071501, 13pp. doi: 10.1063/1.4955026.  Google Scholar

[10]

Y.-P. Choi and J. Lee, Global existence of weak and strong solutions to Cucker-Smale-Navier-Stokes equations in $ \mathbb R^2$, Nonlinear Analysis: Real World Applications, 27 (2016), 158-182.  doi: 10.1016/j.nonrwa.2015.07.013.  Google Scholar

[11]

Y.-P. Choi and B. Kwon, Global well-posedness and large-time behavior for the inhomogeneous Vlasov-Navier-Stokes equations, Nonlinearity, 28 (2015), 3309-3336.  doi: 10.1088/0951-7715/28/9/3309.  Google Scholar

[12]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[13]

Y.-P. ChoiS.-Y. HaJ. Jung and J. Kim, Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids, Nonlinearity, 32 (2019), 1597-1640.  doi: 10.1088/1361-6544/aafaae.  Google Scholar

[14]

Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim, On the coupling of kinetic thermomechanical Cucker-Smale equation and compressible viscous fluid system, J. Math. Fluid Mech., 22 (2020), Art. 4, 34 pp. doi: 10.1007/s00021-019-0466-x.  Google Scholar

[15]

J.-G. DongS.-Y. HaD. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.  Google Scholar

[16]

K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations, Japan J. Indust. Appl. Math., 15 (1998), 51-74.  doi: 10.1007/BF03167396.  Google Scholar

[17]

S.-Y. HaJ. KimC. H. MinT. Ruggeri and X. Zhang, A global existence of classical solutions to the hydrodynamic Cucker-Smale model in presence of a temperature field, Anal. Appl., 16 (2018), 757-805.  doi: 10.1142/S0219530518500033.  Google Scholar

[18]

S.-Y. HaJ. KimC. H. MinT. Ruggeri and X. Zhang, Uniform stability and mean-field limit of a thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.  Google Scholar

[19]

S.-Y. HaJ. Kim and T. Ruggeri, Emergent behaviors of thermodynamic Cucker–Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.  doi: 10.1137/17M111064X.  Google Scholar

[20]

S.-Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal., 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.  Google Scholar

[21]

F. LiY. Mu and D. Wang, Strong solutions to the compressible Navier-stokes-Vlasov-Fokker-Plank equations: global existence near the equilibrium and large time behavior, SIAM J. Math. Anal., 49 (2017), 984-1026.  doi: 10.1137/15M1053049.  Google Scholar

[22]

J. Mathiaud, Local smooth solutions of a thin spray model with collisions, Math. Models Methods Appl. Sci., 20 (2010), 191-221.  doi: 10.1142/S0218202510004192.  Google Scholar

[23]

A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Plank/Navier-Stokes system of equations,, Math. Models Methods Appl. Sci., 17 (2007), 1039-1063.  doi: 10.1142/S0218202507002194.  Google Scholar

[24]

P. O'Rourke, Collective Drop Effects on Vaporizing Liquid Sprays, Princeton University, Princeton, NJ, 1981. Google Scholar

[25]

W. E. Ranz and W. R. Marshall, Evaporization from drops, part Ⅰ-Ⅱ, Chem. Eng. Prog., 48 (1952), 141-180.   Google Scholar

[26]

I. VinkovicC. AguirreS. Simoëns and M. Gorokhovski, Large eddy simulation of droplet dispersion for inhomogeneous turbulent wall flow, Int. J. Multiph. Flow, 32 (2006), 344-364.   Google Scholar

[27]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[28]

D. Wang and C. Yu, Global weak solutions to the inhomogeneous Navier-Stokes-Vlasov equations, J. Differential Equations, 259 (2014), 3976-4008.  doi: 10.1016/j.jde.2015.05.016.  Google Scholar

[29]

C. Yu, Global weak solutions to the incompressible Navier-Stokes-Vlasov equations, J. Math. Pures Appl., 100 (2013), 275-293.  doi: 10.1016/j.matpur.2013.01.001.  Google Scholar

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